Although our results concern discrete singular hyperbolic maps, historically many results about singular hyperbolic attractors come from investigations into hyperbolic flows, the most famous being the flow generated by the Lorenz equations. In [1]}, J. Kaplan and J. Yorke used a Poincaré return map to study the dynamical behavior of the Lorenz attractor, such as the parameters for which periodic points are dense. This Poincaré map was later reformulated as the geometric Lorenz attractor, which is a simplified discrete model of the Poincaré map of the original Lorenz flow. The more general family of discrete Lorenz-type maps was introduced in [2]}. In the years that followed, the Lorenz system and related hyperbolic flows have led to active research in singular hyperbolic attractors (see e.g. [2]}, [4]}, [5]}, [6]}, [7]}, and others). There is also a large body of work on singular hyperbolic and sectional-hyperbolic flows more generally. In [7]}, it is shown that singular hyperbolic flows admit finitely many ergodic physical measures; more recently, it was shown in [9]} that flows of Hölder-\(C^1\)  vector fields admitting a sectional-hyperbolic attracting set admit finitely many ergodic SRB measures. The proof in [9]} also relies on Poincaré return maps, and so these results extend to discrete singular hyperbolic maps arising as Poincaré maps of hyperbolic flows. For a detailed discussion of the ergodic properties of hyperbolic flows and their attractors, see [11]}.
